Gravitational waves from the inspiral of a compact object into a massive, axisymmetric body with arbitrary multipole moments

Abstract

The gravitational waves, emitted by a compact object orbiting a much more massive central body, depend on the central body's spacetime geometry. This paper is a first attempt to explore that dependence. For simplicity, the central body is assumed to be stationary, axially symmetric (but rotating), and reflection symmetric through an equatorial plane, so its (vacuum) spacetime geometry is fully characterized by two families of scalar multipole moments ${\mathit{M}}_{\mathit{l}}$ and ${\mathit{S}}_{\mathit{l}}$ with l=0,1,2,3,..., and it is assumed not to absorb any orbital energy (e.g., via waves going down a horizon or via tidal heating). Also for simplicity, the orbit is assumed to lie in the body's equatorial plane and to be circular, except for a gradual shrinkage due to radiative energy loss. For this idealized situation, it is shown that several features of the emitted waves carry, encoded within themselves, the values of all the body's multipole moments ${\mathit{M}}_{\mathit{l}}$, ${\mathit{S}}_{\mathit{l}}$ (and thus, also the details of its full spacetime geometry). In particular, the body's moments are encoded in the time evolution of the waves' phase \ensuremath{\Phi}(t) (the quantity that can be measurd with extremely high accuracy by interferometric gravitational-wave detectors), and they are also encoded in the gravitational-wave spectrum \ensuremath{\Delta}E(f) (energy emitted per unit logarithmic frequency interval). If the orbit is slightly elliptical, the moments are also encoded in the evolution of its periastron precession frequency as a function of wave frequency, ${\mathrm{\ensuremath{\Omega}}}_{\mathrm{\ensuremath{\rho}}}$(f); if the orbit is slightly inclined to the body's equatorial plane, then they are encoded in its inclinational precession frequency as a function of wave frequency, ${\mathrm{\ensuremath{\Omega}}}_{\mathit{z}}$(f). Explicit algorithms are derived for deducing the moments from \ensuremath{\Delta}E(f), ${\mathrm{\ensuremath{\Omega}}}_{\mathrm{\ensuremath{\rho}}}$(f), and ${\mathrm{\ensuremath{\Omega}}}_{\mathit{z}}$(f). However, to deduce the moments explicitly from the (more accurately measurable) phase evolution \ensuremath{\Phi}(t) will require a very difficult, explicit analysis of the wave generation process---a task far beyond the scope of this paper.